basic_engineering_mathematics0

Areas of plane figures 171
14. Calculate the area of the steel plate shown in
Fig. 22.17.
Dimensions
in mm
25
25
100
A hexagon is a 6-sided polygon which may be divided into 6
equal triangles as shown in Fig. 22.19. The angle subtended at
the centre of each triangle is 360 ◦ /6 = 60 ◦ . The other two angles
in the triangle add up to 120 ◦ and are equal to each other. Hence
each of the triangles is equilateral with each angle 60 ◦ and each
side 8 cm.
h
4cm
8cm
25
60
Fig. 22.17
140
22.4 Further worked problems on areas
of plane figures
Problem 10. Calculate the area of a regular octagon, if
each side is 5 cm and the width across the flats is 12 cm.
An octagon is an 8-sided polygon. If radii are drawn from the
centre of the polygon to the vertices then 8 equal triangles are
produced (see Fig. 22.18).
Fig. 22.19
60°
8cm
Area of one triangle = 1 2 × base × height = 1 2 × 8 × h.
h is calculated using Pythagoras’ theorem:
8 2 = h 2 + 4 2
from which h = √ 8 2 − 4 2 = 6.928 cm
Hence area of one triangle = 1 × 8 × 6.928 = 27.71 cm2
2
Area of hexagon = 6 × 27.71 = 166.3cm 2
Problem 12. Figure 22.20 shows a plan of a floor of a
building which is to be carpeted. Calculate the area of the
floor in square metres. Calculate the cost, correct to the
nearest pound, of carpeting the floor with carpet costing
£16.80 per m 2 , assuming 30% extra carpet is required due
to wastage in fitting.
Fig. 22.18
12 cm
5cm
Area of one triangle = 1 × base × height
2
= 1 2 × 5 × 12
2
= 15 cm2
Area of octagon = 8 × 15 = 120 cm 2
Problem 11. Determine the area of a regular hexagon
which has sides 8 cm long.
2.5 m
L
M
2m
K
4m
0.6 m
J
A
l
0.6 m
H
30°
0.8 m B′ 60°
2m
F
C
0.8 m G
E
D
2m 3m
Fig. 22.20
Area of floor plan
= area of triangle ABC + area of semicircle
+ area of rectangle CGLM + area of rectangle CDEF
− area of trapezium HIJK
3m
3m
B